Design of experiments for the stochastic unit commitment with economic dispatch models

This paper presents a progressive hedging (PH) based decomposition algorithm to improve the computational efficiency of stochastic day-ahead reliability unit commitment (RUC). Stochastic programming is introduced into RUC to facilitate a better decision making for market operation efficiency and energy supply reliability. However, the computational burden of stochastic RUC using today's computing power is still significant. We propose to apply progressive hedging (PH) algorithm to decompose the stochastic RUC problem. It is shown that a PH-based algorithm allows parallel computation of stochastic RUC, which can be much faster than conventional stochastic programming methods. A practical power system is used to verify the effectiveness of the proposed computation framework.

Owing to the massive deployment of renewablepower production units over the last couple of decades, the useof stochastic optimization methods to solve the unit commitmentproblem has gained increasing attention. Solving stochastic unitcommitment problems in large-scale power systems requires high computational power, as stochastic models are dramaticallymore complex than their deterministic counterparts. This paperprovides new insight into the potential of Progressive Hedgingto decrease the solution time of the stochastic unit commitmentproblem with a relatively small trade-off in terms of thesuboptimality of the solution. Computational studies show thatthe run-time is at most half of what is needed to solve theoriginal extensive formulation of the problem, when more thanten wind power scenarios are utilized. These studies demonstrategreat potential for solving real-world stochastic unit commitmentproblems using the Progressive Hedging algorithm.

As an attempt to reduce reliance on fossil fuels to satisfy electricity demand, the penetration of renewable sources like wind or solar power has experienced rapid growth in recent years. Due to their intermittent nature, the exact contribution of renewable sources to electricity supply is at least partly unknown. In order to be able to accommodate this uncertainty in real-time, sufficient capacities in the form of thermal and hydro power plants must be available. On that account, power system operators solve the so-called unit commitment problem after bidding in the day-ahead market closes to derive schedules for their power plants which can ensure system reliability at low costs. In literature, two approaches have mainly been studied in this context: Robust and stochastic unit commitment. Being a worst-case formulation, robust unit commitment puts its focus on reliability. A common drawback of this approach is that it tends to deliver over-conservative schedules. Stochastic unit commitment on the other hand creates more cost-effective schedules by preparing for the expected case, but either fails to guarantee system reliability or puts a high workload on the CPU. The limitations of known formulations have sparked interest in so-called hybrid approaches, which aim at combining the ideas of robust and stochastic unit commitment in a favorable way. This paper reports a novel hybrid approach to solve the unit commitment under uncertainty, which yields both robust and cost-efficient schedules. The new method respects the continuous nature of uncertainties and is thus in particular favorable for applications in power systems with high penetration of volatile renewable sources. By merging the ideas of robust and stochastic unit commitment, the proposed hybrid formulation minimizes the expected worst-case dispatch costs. Our method relies on partitioning the continuous range of the uncertainties into subsets. By means of the number of partitions, the solution can be adjusted between the conservative robust and the cost-efficient stochastic unit commitment in a user-friendly manner. A Benders decomposition algorithm is derived to solve the hybrid unit commitment efficiently. Finally, a case study confirms the superior performance of the proposed method.

The increasing number of gas-fired units has significantly intensified the coupling between electric and gas power networks. Traditionally, nonlinearity and nonconvexity in gas flow equations, together with renewable-induced stochasticity, resulted in a computationally expensive model for unit commitment in electricity-gas coupled integrated energy systems (IES). To accelerate stochastic day-ahead scheduling, we applied and modified Progressive Hedging (PH), a heuristic approach that can be computed in parallel to yield scenario-independent unit commitment. Through early termination and enumeration techniques, the modified PH algorithm saves considerable computational time for certain generation cost settings or when the scale of the IES is large. Moreover, an adapted second-order cone relaxation (SOCR) is utilized to tackle the nonconvex gas flow equation. Case studies were performed on the IEEE 24-bus system/Belgium 20-node gas system and the IEEE 118-bus system/Belgium 20-node gas system. The computational efficiency when employing PH is 188 times that of commercial software, and the algorithm even outperforms Benders Decomposition. At the same time, the gap between the PH algorithm and the benchmark is less than 0.01% in both IES systems, which proves that the solutions produced by PH reach acceptable optimality in this stochastic UC problem.

The increasing penetration of renewable generation poses challenges to grid operation, including the need for increased available transfer capability (ATC). The power flow control capability of flexible AC transmission system (FACTS) devices can enhance the transfer capability over the existing network, thus improving the utilization of renewable energy. To harness such benefits, it is essential to integrate FACTS operation into stochastic unit commitment (SUC), in which renewable generation uncertainties are explicitly modeled. Furthermore, to ensure the computational tractability of SUC with FACTS incorporated, it is vital to study the impacts of FACTS deployment on computational burden of the model. This paper proposes a SUC model with the consideration of optimal FACTS operation. The proposed model is solved using the progressive hedging (PH) algorithm. PH is also used to solve the original SUC model for comparison. Simulation results reveal that despite the increased computational complexity due to FACTS operation, PH is a valuable approach in ensuring that the proposed model can be solved efficiently.

We present a distributed asynchronous algorithm for solving two-stage stochastic mixed-integer programs (SMIP) using scenario decomposition, aimed at industrial-scale instances of the stochastic unit commitment (SUC) problem. The algorithm is motivated by large differences in run times observed among scenario subproblems of SUC instances, which can result in inefficient use of distributed computing resources by synchronous parallel algorithms. Our algorithm performs dual iterations asynchronously using a block-coordinate subgradient descent method which allows performing block-coordinate updates using delayed information, while candidate primal solutions are recovered from the solutions of scenario subproblems using heuristics. We present a high performance computing implementation of the asynchronous algorithm, detailing the operations performed by each parallel process and the communication mechanisms among them. We conduct numerical experiments using SUC instances of the Western Electricity Coordinating Council system with up to 1000 scenarios and of the Central Western European system with up to 120 scenarios. We also conduct numerical experiments on generic SMIP instances from the SIPLIB library (DCAP and SSLP). The results demonstrate the general applicability of the proposed algorithm and its ability to solve industrial-scale SUC instances within operationally acceptable time frames. Moreover, we find that an equivalent synchronous parallel algorithm would leave cores idle up to 80.4% of the time on our realistic test instances, an observation which underscores the need for designing asynchronous optimization schemes in order to fully exploit distributed computing on real world applications.

Integration of renewable generation uncertainties into stochastic unit commitment considering reserve and risk: A comparative study

Unit Commitment (UC) is an essential decision taken by ISO to ensure reliable and economic operation of the power system. Renewable energy penetration is increasing in a power system due its inexhaustive and environment friendly nature. Renewable energy poses some challenges to the present power systems due to its high uncertainty. In addition to this, the impact of contingencies on the power system also cannot be ignored. Both renewable energy and contingencies make the power system more prone to uncertain behaviour which can affect the economics as well as reliability of the power system. In this paper, UC is solved using deterministic and stochastic approaches. Deterministic UC considers forecasted wind data and Stochastic UC considers uncertain scenarios of the wind. Practically the power system is more prone to uncertainty because of renewable energy penetration and contingencies. Thus, this paper discusses how a UC decision varies as more uncertainties arises. Mixed Integer Linear Programming (MILP) is used for obtaining the UC decision results for IEEE 30-bus system. The results of UC, power dispatch and generation costs obtained with the two approaches have been compared.

As more wind power and other renewable resources are being integrated into the electric power grid, the forecast uncertainty brings operational challenges for the power system operators. In this report, different operational strategies for uncertainty management are presented and evaluated. A comprehensive and consistent simulation framework is developed to analyze the performance of different reserve policies and scheduling techniques under uncertainty in wind power. Numerical simulations are conducted on a modified version of the IEEE 118-bus system with a 20% wind penetration level, comparing deterministic, interval, and stochastic unit commitment strategies. The results show that stochastic unit commitment provides a reliable schedule without large increases in operational costs. Moreover, decomposition techniques, such as load shift factor and Benders decomposition, can help in overcoming the computational obstacles to stochastic unit commitment and enable the use of a larger scenario set to represent forecast uncertainty. In contrast, deterministic and interval unit commitment tend to give higher system costs as more reserves are being scheduled to address forecast uncertainty. However, these approaches require a much lower computational effort Choosing a proper lower bound for the forecast uncertainty is important for balancing reliability and system operational cost in deterministic and interval unit commitment. Finally, we find that the introduction of zonal reserve requirements improves reliability, but at the expense of higher operational costs.

This paper focuses on the optimal unit commitment (UC) scheme along with optimal power trading for the Northeast Power System (NEPS) of Afghanistan with a penetration of 230 MW of PV power energy. The NEPS is the biggest power system of Afghanistan fed from three main sources; 1. Afghanistan’s own power generation units (three thermal units and three hydro units); 2. imported power from Tajikistan; 3. imported power from Uzbekistan. PV power forecasting fluctuations have been handled by means of 50 scenarios generated by Latin-hypercube sampling (LHS) after getting the point solar radiation forecast through the neural network (NN) toolbox. To carry out the analysis, we consider three deterministic UC and two stochastic UC cases with a two-stage programming model that indicates the day-ahead UC as the first stage and the intra-day operation of the system as the second stage. A binary-real genetic algorithm is coded in MATLAB software to optimize the proposed cases in terms of thermal units’ operation costs, import power tariffs, as well as from the perspective of the system reliability risks expressed as the reserve and load not served costs. The results indicate that in the deterministic UC models, the risk of reserve and load curtailment does exist. The stochastic UC approaches including the optimal power trading are superior to the deterministic ones. Moreover, the scheduled UC costs and reserves are different from the actual ones.

In recent years, high penetration of variable generating sources, such as wind power, has challenged independent system operators (ISO) in keeping a cheap and reliable power system operation. Any deviation between expected and real wind production must be absorbed by the power system resources (reserves), which must be available and ready to be deployed in real time. To guarantee this resource availability, the system resources must be committed in advance, usually the day-ahead, by solving the so-called unit commitment (UC) problem. If the quantity of committed resources is extremely low, there will be devastating and costly consequences in the system, such as significant load shedding. On the other hand, if this quantity is extremely high, the system operation will be excessively expensive, mainly because facilities will not be fully exploited. This thesis proposes computationally efficient models for optimal day-ahead planning in (thermal) power systems to adequately face the stochastic nature of wind production in the real-time system operation. The models can support ISOs to face the new challenges in short-term planning as uncertainty increases dramatically due to the integration of variable generating resources. This thesis then tackles the UC problem in the following aspects: Power system representation: This thesis identifies drawbacks of the traditional energy-block scheduling approach, which make it unable to adequately prepare the power system to face deterministic and perfectly known events. To overcome those drawbacks, we propose the ramp-based scheduling approach that more accurately describes the system operation, thus better exploiting the system flexibility. UC computational performance: Developing more accurate models would be pointless if these models considerably increase the computational burden of the UC problem, which is already a complex integer and non-convex problem. We then devise simultaneously tight and compact formulations under the mixed-integer programming (MIP) approach. This simultaneous characteristic reinforces the convergence speed by reducing the search space (tightness) and simultaneously increasing the searching speed (compactness) with which solvers explore that reduced space. Uncertainty management in UC: By putting together the improvements in the previous two aspects, this thesis contributes to a better management of wind uncertainty in UC, even though these two aspects are in conflict and improving one often means harming the other. If compared with a traditional energy-block UC model under the stochastic (deterministic) paradigm, a stochastic (deterministic) ramp-based UC model: 1) leads to more economic operation, due to a better and more detailed system representation, while 2) being solved significantly faster, because the core of the model is built upon simultaneously tight and compact MIP formulations. To further improve the uncertainty management in the proposed ramp-based UC, we extend the formulation to a network-constrained UC with robust reserve modelling. Based on robust optimization insights, the UC solution guarantees feasibility for any realization of the uncertain wind production, within the considered uncertainty ranges. This final model remains as a pure linear MIP problem whose size does not depend on the uncertainty representation, thus avoiding the inherent computational complications of the stochastic and robust UCs commonly found in the literature.

Managing solar uncertainty in microgrid systems with stochastic unit commitment

In this dissertation we develop numerical algorithms and software tools to facilitate parallel solutions of nonlinear programming (NLP) problems. In particular, we address large-scale, block-structured problems with an intrinsic decomposable configuration. These problems arise in a great number of engineering applications, including parameter estimation, optimal control, network optimization, and stochastic programming. The structure of these problems can be leveraged by optimization solvers to accelerate solutions and overcome memory limitations, and we propose variants to two classes of optimization algorithms: augmented Lagrangian (AL) schemes and Schur-complement interior-point methods. The convergence properties of augmented Lagrangian decomposition schemes like the alternating direction method of multipliers (ADMM) and progressive hedging (PH) are well established for convex optimization but convergence guarantees in non-convex settings are still poorly understood. In practice, however, ADMM and PH often perform satisfactorily in complex non-convex NLPs. In this work, we study connections between the method of multipliers (MM), ADMM, and PH to derive benchmarking metrics that explain why PH and ADMM work in practice. We illustrate the concepts using challenging dynamic optimization problems. Our exposition seeks to establish more formalism in benchmarking ADMM, PH, and AL schemes and to motivate algorithmic improvements. The effectiveness of nonlinear interior-point solvers for solving large-scale problems relies quite heavily on the solution of the underlying linear algebra systems. The schur-complement decomposition is very effective for parallelizing the solution of linear systems with modest coupling. However, for systems with large number of coupling variables the schur-complement method does not scale favorably. We implement an approach that uses a Krylov solver (GMRES) preconditioned with ADMM to solve block-structured linear systems that arise in the interior-point method. We show that this ADMM-GMRES approach overcomes the well-known scalability issues of Schur decomposition. One important drawback of using decomposition approaches like ADMM and PH is their convergence rate. Unlike Schur-complement interior-point algorithms that have super-linear convergence, augmented Lagrangian approaches typically exhibit linear and sublinear rates. We exploit connections between ADMM and the Schur-complement decomposition to derive an accelerated version of ADMM. Specifically, we study the effectiveness of performing a Newton-Raphson algorithm to compute multiplier estimates for augmented Lagrangian methods. We demonstrate using two-stage stochastic programming problems that our multiplier update achieves convergence in fewer iterations for MM on general nonlinear problems. In the case of ADMM, the newton update significantly reduces the number of subproblem solves for convex quadratic programs (QPs). Moreover, we show that using newton multiplier updates makes the method robust to the selection of the penalty parameter. Traditionally, state-of-the-art optimization solvers are implemented in low-level programming languages. In our experience, the development of decomposition algorithms in these frameworks is challenging. They present a steep learning curve and can slow the development and testing of new numerical algorithms. To mitigate these challenges, we developed PyNumero, a new open source framework implemented in Python and C++. The package seeks to facilitate development of optimization algorithms for large-scale optimization within a high-level programming environment while at the same time minimizing the computational burden of using Python. The efficiency of PyNumero is illustrated by implementing algorithms for problems arising in stochastic programming and optimal control. Timing results are presented for both serial and parallel implementations. Our computational studies demonstrate that with the appropriate balance between compiled code and Python, efficient implementations of optimization algorithms are achievable in these high-level languages.

Solar cell generators (SCGs) offer a renewable and environmentally-friendly source for generating electricity. However, their intermittent nature affects power system reliability. This paper studies the effect of SCGs on power system reliability under various penetration levels. A unit commitment economic dispatch (UCED) model is developed to compute the expected generation output by taking into account normal and contingency operations of conventional generators (CGs) and the forecasted solar radiation and forced outage rates (FORs) of solar farms (SFs). The UCED is applied on a test system consisting of 10 CGs and different penetration levels of solar generation. Then, the expected SCGs power output and the forecasted load probabilistic model is inputted to the probabilistic production costing (PPC) method to assess reliability, compute projected production cost, and capacity credit for the SCGs.

Independent System Operators (ISOs) worldwide face the ever-increasing challenge of coping with uncertainties, which requires sophisticated algorithms for solving unit-commitment (UC) problems of increasing complexity in less-and-less time. Hence, decomposition methods are appealing options to produce easier-to-handle problems that can hopefully return good solutions at reasonable times. When applied to two-stage stochastic models, decomposition often yields subproblems that are embarrassingly parallel. Synchronous parallel-computing techniques are applied to the decomposable subproblem and frequently result in considerable time savings. However, due to the inherent run-time differences amongst the subproblem’s optimization models, unequal equipment, and communication overheads, synchronous approaches may underuse the computing resources. Consequently, asynchronous computing constitutes a natural enhancement to existing methods. In this work, we propose a novel extension of the asynchronous level decomposition to solve stochastic hydrothermal UC problems with mixed-integer variables in the first stage. In addition, we combine this novel method with an efficient task allocation to yield an innovative algorithm that far outperforms the current state-of-the-art. We provide convergence analysis of our proposal and assess its computational performance on a testbed consisting of 54 problems from a 46-bus system. Results show that our asynchronous algorithm outperforms its synchronous counterpart in terms of wall-clock computing time in 40% of the problems, providing time savings averaging about 45%, while also reducing the standard deviation of running times over the testbed in the order of 25%.